Here, it is clear that Hamilton was the first to introduce the del operator, but that Tait was the one to rotate Hamilton's symbol 90 degrees into its modern shape.
Some uncertainty exists whether it was Tait or Maxwell who dubbed it "nabla":

VECTOR CALCULUS SYMBOLS
For vector analysis entries on the Words pages, see here for a list.

The vector differential operator, now written and called nabla or del, was introduced by William Rowan Hamilton (1805-1865). Hamilton wrote the operator as and it was P. G. Tait who established as the conventional symbol--see his An Elementary Treatise on Quaternions (1867). Tait was also responsible for establishing the term nabla. See NABLA on the Earliest Uses of Words page.

David Wilkins suggests that Hamilton may have used the nabla as a general purpose symbol or abbreviation for whatever operator he wanted to introduce at any time. In 1837 Hamilton used the nabla, in its modern orientation, as a symbol for any arbitrary function in Trans. R. Irish Acad. XVII. 236. (OED.) He used the nabla to signify a permutation operator in "On the Argument of Abel, respecting the Impossibility of expressing a Root of any General Equation above the Fourth Degree, by any finite Combination of Radicals and Rational Functions," Transactions of the Royal Irish Academy, 18 (1839), pp. 171-259.

Hamilton used the rotated nabla, i.e. , for the vector differential operator in the "Proceedings of the Royal Irish Academy" for the meeting of July 20, 1846. This paper appeared in volume 3 (1847), pp. 273-292. Hamilton also used the rotated nabla as the vector differential operator in "On Quaternions; or on a new System of Imaginaries in Algebra"; which he published in instalments in the Philosophical Magazine between 1844 and 1850. The relevant portion of the paper consists of articles 49-50, in the instalment which appeared in October 1847 in volume 31 (3rd series, 1847) of the Philosophical Magazine, pp. 278-283. A footnote in vol. 31, page 291, reads:
......
In that paper designed for Southampton the characteristic was written ; but this more common sign has been so often used with other meanings, that it seems desirable to abstain from appropriating it to the new signification here proposed.
Wilkins writes that "that paper" refers to an unpublished paper that Hamilton had prepared for a meeting of the British Association for the Advancement of Science, but which had been forwarded by mistake to Sir John Herschel's home address, not to the meeting itself in Southampton, and which therefore was not communicated at that meeting. The footnote indicates that Hamilton had originally intended to use the nabla symbol that is used today but then decided to rotate it to avoid confusion with other uses of the symbol.
The rotated form appears in Hamilton's magnum opus, the Lectures on Quaternions (1853, p. 610).